PHYS 580

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PHYS 580

• April 12, 2005

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Homework 6, Question 2 To achieve a photometric
accuracy of 0m.05, the radiant flux must be
measured to an accuracy of what percent? ? 4.5
(or 4.7, depending) SADE Proposal Optical
telescope designed to achieve 0.01
precision. How do we apply these to
observations? E.g., based on your findings in
the homework, we must ensure that when we finish
for the night, and put the telescope away, our
flux measurement is known to within 4.5, so that
we have measured F ? ?F where ?F/F ? 0.045
, And this will yield a magnitude, ? 0.05 .
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• For the SADE proposal we discussed last week,
  recall that we wanted 0.01 uncertainty in the
  visible-light observations.
• First, how did we arrive at that criterion?
• Second, how did we turn that into an
  observational constraint?
• Finally, in designing the scope, how big do we
  have to make the primary mirror?

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• For the SADE proposal we discussed last week,
  recall that we wanted 0.01 uncertainty in the
  visible-light observations.
• First, how did we arrive at that criterion?

We estimated the amount of sunlight that would be
blocked to an extraterrestrial observer when the
Earth passes in front of the Sun. Also, we
consulted big magic.
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• For the SADE proposal we discussed last week,
  recall that we wanted 0.01 uncertainty in the
  visible-light observations.
• First, how did we arrive at that criterion?
• Second, how did we turn that into an
  observational constraint?

We defined the precision as based on photon
counting statistics, whereby ?N/N
1/?N Therefore, 0.01 ? 10-4 ? 1/?108 , so
that N 108 photons.
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• For the SADE proposal we discussed last week,
  recall that we wanted 0.01 uncertainty in the
  visible-light observations.
• First, how did we arrive at that criterion?
• Second, how did we turn that into an
  observational constraint?
• Finally, in designing the scope, how big do we
  have to make the primary mirror?

With V 0, get 104 photons cm-2 s-1 nm-1 . With
a bandwidth of, say 100 nm, get 106 photons cm-2
s-1 . With V 7, get 1580 photons cm-2 s-1 . If
you want to complete the observation in no more
than 600 seconds, then how much collecting area
do you need? ? 5 inches diameter.
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BE CAREFUL! Are you really counting photons, or
are you monitoring current? Be very specific in
your definitions. In a narrow-band instrument,
the photon-counting assumption is more
forgivable, if not more accurate, because each of
the photons really does have nearly the same
energy. And if you really do count each and
every photon, then
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Example Pretend your mission design is required
to produce 10 precisions in all measurements.
What does that say to you?

• ?N/N 0.1 ? N 100 , so all you need is 100
  photons
• Thats 100 photons per pixel, because each pixel
  is an independent measurement.
• For AIA, the design spec further limits us to
  maximum exposures of 2.7 seconds, so we must
  design for
• 100 photons per pixel per (2.7 seconds) 37
  photons pixel-1 s-1 .
• Knowing that the target produces S? photons
  cm-2 s-1 sr-1 in the ? passband, we can design
  the area and plate scale of the telescope
  accordingly.

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But what if we wish to combine observations to
derive a result? We still want to have 10
precision there too. Now we have to think about
combining uncertainties. Two relationships you
should know If C A B, or C A B,
then ?c2 ?a2 ?b2 If C A/B, or C A ? B,
then (?c/C)2 (?a/A)2 (?b/B)2
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(2) Now, rightly or wrongly, a common thing to do
with the signals is to make ratios, so that
Result (signal A) ? (signal B) If we assume
that (signal A) is roughly equal to (signal B),
then for ?Result/Result 0.01 We need signal A
? signal B 200 photons/pixel If (signal A) and
(signal B) are vastly different, then the math is
not much harder.
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(3) One more scenario. Sometimes its nice to
compare images from different times and try to
get a speed. Itll be an apparent speed in the
plane of the sky, and its really only done in
solar work, whereas the previous two applications
can be adapted to nearly anything. Lets say a
wavefront is moving across the field of view with
speed v . During an exposure, which lasts for a
duration of texp , the position of the wavefront
is blurred by ?x v texp .